Whakaoti i te 9x^2+150x-119=0

$$9 { x }^{ 2 } +150x-119=0$$

Whakaoti mō x

$$x=\frac{2\sqrt{186}-25}{3}\approx 0.758787798$$

$$x=\frac{-2\sqrt{186}-25}{3}\approx -17.425454465$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$9x^{2}+150x-119=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, 150 mō b, me -119 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-150±\sqrt{150^{2}-4\times 9\left(-119\right)}}{2\times 9}$$

Pūrua 150.

$$x=\frac{-150±\sqrt{22500-4\times 9\left(-119\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-150±\sqrt{22500-36\left(-119\right)}}{2\times 9}$$

Whakareatia -36 ki te -119.

$$x=\frac{-150±\sqrt{22500+4284}}{2\times 9}$$

Tāpiri 22500 ki te 4284.

$$x=\frac{-150±\sqrt{26784}}{2\times 9}$$

Tuhia te pūtakerua o te 26784.

$$x=\frac{-150±12\sqrt{186}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{-150±12\sqrt{186}}{18}$$

Nā, me whakaoti te whārite x=\frac{-150±12\sqrt{186}}{18} ina he tāpiri te ±. Tāpiri -150 ki te 12\sqrt{186}.

$$x=\frac{12\sqrt{186}-150}{18}$$

Whakawehe -150+12\sqrt{186} ki te 18.

$$x=\frac{2\sqrt{186}-25}{3}$$

Nā, me whakaoti te whārite x=\frac{-150±12\sqrt{186}}{18} ina he tango te ±. Tango 12\sqrt{186} mai i -150.

$$x=\frac{-12\sqrt{186}-150}{18}$$

Whakawehe -150-12\sqrt{186} ki te 18.

$$x=\frac{-2\sqrt{186}-25}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{186}-25}{3} x=\frac{-2\sqrt{186}-25}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$9x^{2}+150x-119=0$$

Me tāpiri 119 ki ngā taha e rua o te whārite.

$$9x^{2}+150x-119-\left(-119\right)=-\left(-119\right)$$

Mā te tango i te -119 i a ia ake anō ka toe ko te 0.

$$9x^{2}+150x=-\left(-119\right)$$

Tango -119 mai i 0.

$$9x^{2}+150x=119$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}+150x}{9}=\frac{119}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}+\frac{150}{9}x=\frac{119}{9}$$

Whakahekea te hautanga \frac{150}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{50}{3}x=\frac{119}{9}$$

Whakawehea te \frac{50}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{25}{3}. Nā, tāpiria te pūrua o te \frac{25}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{50}{3}x+\left(\frac{25}{3}\right)^{2}=\frac{119}{9}+\left(\frac{25}{3}\right)^{2}$$

Pūruatia \frac{25}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{50}{3}x+\frac{625}{9}=\frac{119+625}{9}$$

Tāpiri \frac{119}{9} ki te \frac{625}{9} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+\frac{50}{3}x+\frac{625}{9}=\frac{248}{3}$$

Tauwehea x^{2}+\frac{50}{3}x+\frac{625}{9}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{25}{3}\right)^{2}=\frac{248}{3}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x+\frac{25}{3}\right)^{2}}=\sqrt{\frac{248}{3}}$$

Whakarūnātia.

$$x+\frac{25}{3}=\frac{2\sqrt{186}}{3} x+\frac{25}{3}=-\frac{2\sqrt{186}}{3}$$

Me tango \frac{25}{3} mai i ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{186}-25}{3} x=\frac{-2\sqrt{186}-25}{3}$$